Results of Multivariate Regression


The results of the multivariate regression modeling showed that the three variables—lake elevation, Williamson River inflow, and wind speed—were sufficient to explain most of the variability in the computed total and partial replacement rates (table 3, fig. 12). The amount of variability explained by the best regression for each variable, as measured by the coefficient of determination R², ranged from 87 percent for RR-TUL to 97 percent for RR-TULwWR, and all regressions were significant (p<0.0001). The inclusion of east-west and north-south wind components rather than wind speed alone as independent variables improved the fit of the regression for three out of four dependent variables (table 3). Thus, the incorporation of wind direction generally resulted in a better model for the replacement rates—the improvement in fit as measured by R² ranged from 3.4 percent for RR-GB to 7.8 percent for RR-TULwWR. The R² values associated with the regression models based on wind speed alone, however, were all 0.87 or greater (table 3); therefore, depending on the question being addressed, the greater simplicity of the wind-speed-only models may outweigh the benefit of the better fit obtained by using the models that incorporate wind direction. 


Most regressions were based on 384 tracer experiments, derived from (32 wind values) × (4 Williamson River inflow values) × (3 lake elevation values). The regressions for partial replacement rates in both sides of the Delta (RR-GBwWR and RR-TULwWR) that incorporated the magnitude of wind components as independent variables have a lower N and a higher valid minimum lake elevation than all the other regressions, because the tracer experiments at the lowest lake elevation were not used in developing those regressions. A satisfactory regression model for those dependent variables, based on using wind components as independent variables (determined by an R²>0.80), could not be developed without removing the tracer experiments at the lowest elevation, a consequence of the fact that exchange between the Williamson River and Goose Bay or Tulana becomes severely constrained and relatively insensitive to wind direction at an unknown elevation between 4,140.5 and 4,141.5 ft (fig. 11). Satisfactory regressions using wind speed as an independent variable could be developed for RR-GBwWR and RR-TULwWR when all three lake elevations were included. 


The regression models quantify the dependencies of replacement rates on wind, Williamson River inflow, and lake elevation shown in figures 8–11, and interpolate those relations continuously across the parameter space. When the wind-speed only regression models are presented as contour plots in two-dimensional parameter space, the increase of both RR-GB and RR-TUL with lake elevation, Williamson River inflow, and wind speed (figs. 13 and 14A, B) is clear. RR-TUL has a stronger dependence on wind speed than does RR-GB. It is also apparent that both RR-GBwWR and RR-TULwWR increase with lake elevation and Williamson River inflow, but RR-GBwWR increases with wind speed, whereas RR-TULwWR decreases with wind speed (figs. 13 and 14C and D). Thus, an increase in the strength of the wind forcing at the water surface creates a tradeoff between the two sides of the Delta, such that more Williamson River water flows into the Goose Bay side at higher wind speeds. To illustrate the dependence on wind direction, the regression models that use wind components as independent variables to explain RR-GBwWR and RR-TULwWR are plotted in the parameter space defined by the east-west and north-south wind component (fig. 15). The partial replacement in Goose Bay, RR-GBwWR, increases with the westerly and the northerly components of the wind, whereas the partial replacement in Tulana, RR-TULwWR, has the opposite dependence. Thus, winds with both a northerly and westerly component, which are the prevailing winds during the spring and summer months, result in more movement of Williamson River water into the Goose Bay side of the Delta than into the Tulana side.


First posted March 29, 2012